International Mathematial Forum, Vol. 9, 4, no. 5, 97-5 HIKARI Ltd, www.m-hiari.om http://d.doi.org/.988/imf.4.465 Global Properties of an Improed Hepatitis B Virus Model with Beddington-DeAngelis Infetion Rate and CTL Immune Response Zhengzhi Cao Shool of Statistis and Applied Mathematis, Anhui Uniersity of Finane and Eonomis, Bengbu, China Copyright 4 Zhengzhi Cao. This is an open aess artile distributed under the Creatie Commons Attribution Liense, whih permits unrestrited use, distribution, and reprodution in any medium, proided the original wor is properly ited. Abstrat This paper inestigates the global stability of an improed hepatitis B irus model with Beddington-DeAngelis infetion rate. CTL immune response is studied by onstruting Lyapuno funtions. If the basi prodution number is less than or equal to one, the uninfeted steady state is globally asymptotially stable. If the basi prodution number is more than one, the immune-free equilibrium is globally asymptotially stable. If the immune response reprodution number is more than one, the endemi equilibrium is globally asymptotially stable. Keywords: Virus dynamis, CTL immune response, Beddington-DeAngelis, Lyapuno funtion, Global stability Introdution In the past deades, there has been muh interest in mathematial modeling of HIV dynamis [-]. Clearl we are now able to understand the dynamis of infetions at the ellular leel. Of the many different mehanisms of the immune system, defenses against iral infetions are of interest beause many of the diseases aused by them, e.g. hepatitis B and AIDS, are hroni and inurable
98 Zhengzhi Cao []. To model the immune response during a iral infetion, researhers first onsider the basi interations between the immune system and the irus using the following system of differential equations [, 3]. Then we introdue the model onstruted by Nowa and Bangham [9]. & λ d β y& β ay pyz, & y u z& yz bz, (.) where suseptible ells are produed at a onstant rate λ, die at a density-dependent rate d, and beome infeted with a rate β ; infeted ells y are produed at rate β and die at a density-dependent rate ay ; free irus partiles are released from infeted ells at a rate y and die at a rateu. z is the onentration of CTLs. Infeted ells y are remoed at a rate pz by the CTL immune response and the irus-speifi CTL ells proliferate at a rate y by ontat with infeted ells, and die at a ratebz. In addition, we tae the non ytolyti mehanisms of CTL ells into onsideration, and build the following model: & λ d β qyz, y& β ay ( p q) yz, & y u z& yz bz, (.) the bilinear term qyz represents the CTLells ure the infeted hepatoytes by a nonlyrieffetors mehanism [8]. It is true that the rate of infetion in most irus dynamis models is assumed to be bilinear in the irus and suseptible ells. Howeer, the atual inidene rate is probably not linear oer the entire range of and.thus, it is reasonable to assume that the infetion rateof HIV- is gien by Beddington-DeAngelis β β funtional response,, m, n. The funtional response was introdued by Beddington [] and DeAngelis et al. [4]. In general we inorporate a Beddingto-DeAngelis into the model (.) and onstrut the following model:
Global properties of an improed Hepatitis B irus model 99 β & λ d qyz, β y& ay ( p q) yz, (.3) & y u z& yz bz, here the state ariable, z and parameters λ, d, β, q, a, p,, u,, b hae the same biologial meanings as in the model (.). The initial ondition of (.3) is, y,, z. ( ) ( ) ( ) ( ) Global asymptotial stability Note that the basi reprodutie ratio of the system (.3) is R λβ ( dau auλm). From (.3), we an obtain that (i) if R, then the uninfeted steady state E ( /,,, λ d ) is the unique steady state, alled the infetion-free equilibrium. * (ii) if R >, R, then in addition to the uninfeted steady state, there eists an immune-free equilibrium E, y,, ), here ( z λn au λβ λ β, y, β nd aum a We define an immune response reprodution number R* y b. If R >, the infeted ell for per unit time is λ β ( R ) aβ and a um, in spite of CTL immune response. CTL ells reprodued by infeted ells stimulating per unit time is y. The CTL load of a ell during the life yle is y b. (iii) if R * >, orresponding to the surial of the irus and CTL ells, there E, y,, z, here is an endemi equilibrium ( ) ( ) ( ) β nd aum R au β nd aum R ( d nd β λm) ( d nd β λm) 4md( λ λn ) b y, u dm λ d ay y, z py In this setion, we onsider the global asymptotial stability of these three equilibria. Theorem. The infetion-free equilibrium E is globally asymptotially stable if
Zhengzhi Cao R. Proof. Define a Lyapuno funtion L as follows: a p L (, ln y z. m (.) where λ d, along the positie solutions of model (.3), we alulating the time deriatie of L (,, then d L d t m & m & y& a & p z& β β λ d qyz λ d qyz m m n m m n β a p ay ( p q) yz ( y u) ( yz b m n d au m m n m n Thus, sine and R, we find that L & (,,, y for all ( m) ( ) aun au pbz R qyz., z,the infetion-free equilibrium E is stable. For L & (,, when and z. Set M be the largest inariant set in the set E {(, L& (, } {(,, y,, z }. It is lear that M { E}. The global asymptotial stability of E follows from LaSalle inariane priniple [5]. Theorem.The immune-free equilibrium E is globally asymptotially stable if * R >, R. Proof. Define a Lyapuno funtion L as follows: ay y a p q L y z s y y y s y,,, d ln ln β ms n (.) ( ) z. Along the positie solutions of model (.3), we alulating the time deriatie of L (,, we obtain d L y a p q & y& ay & y& & & z& dt β y Obiousl L (, is positie define with respet to (, y
Global properties of an improed Hepatitis B irus model β λ d ay pyz ay λ d qyz β y β a a p q ay y m n Sine (, is a positie equilibrium point of (.3), we hae λ d a au ay, ay β m n Thus, we obtain au λ d d ay d ay (.3) What s more, β ay λ d qyz β ( d ay d) ay ay qyz m n β y β ay pyz (.4) y Then, we get d L y m n d ay d m n m n y y m n y m n ay ay ay qyz qyz y y β b b qz y qyz ay pz y m n ( p q) yz ( y u) ( y u) ( yz b. ( ) ( ) ( ) d n n( m)( ) ay ( )( ) y y ay 4 m n y m n y. b b qyz pz y qz y (.5) m n Sine the arithmeti mean is greater than or equal to the geometri mean, it is lear that y y 4 y y
Zhengzhi Cao and the equality holds only for, y z z. From R, we hae y b, then pz ( y b ), qz( y b ). Therefore, d L dt holds for all, z >. Thus, the immune-free equilibrium E is stable. And we hae L (, if and only if, y z z and R. The largest ompat inariant set in M {(, L (, } is { E }. Therefore, the immune-free equilibrium E is globally asymptotially stable by the LaSalle inariane priniple [5]. Theorem.3The endemi equilibrium E is globally asymptotially stable if R >. Proof. Define a Lyapuno funtion L 3 as follows: ay ( ) ( p q) y z y L, z d s y y y ln βs y ms n ( p q) p q z a z z z z ln ln. z (.6) Along the positie solutions of model (.3), we alulating the time deriatie of L 3 (,, we obtain d L 3 y a [ ( ) ] ( p q) z & y& ay p q y z & y& & & d t β y ( p q z z& z& z.7) Clearl u y λ d ay py z, β [ ay ( p q) y z ]( ), It follows from (.7), we obtain d L 3 a ( ) ( p q) z λ d ay pyz λ d ( y u) dt a ( p q) z p q p q z ( y u) ( yz b ( yz b z (.8) β [ ay ( p q) y z ] qyz β y y β ay ( p q) yz.
Global properties of an improed Hepatitis B irus model 3 Note that, y β ay y a ( p q) ( p q) a Hene, d L3 d dt z z [ y z ] ( p q) yz ay ( p q) ay ( p q) y z. ( y u) ay ( p q) yz ay ( p q) y z. y y ( y u) ay ( p q) yz ay ( p q) y z. m n m n [ ay ( p q) y z ] y [ ay ( p q) y z ] ay ( p q) ( ay ( p q) y z ) y β qyz. [ y z ] ( ) ( ) ( ) [ ( ) ] ( )( ) d n n m ay p q y z ( )( ) y y [ ay ( p q) y z ] 4 m [ ay ( p q) y z ] qyz. n m n y y y y m n β (.9) Sine the arithmeti mean is greater than or equal to the geometri mean, it is lear that y y 4 y y and the equality holds only for, y y,. It follows from (.9) that L dt holds for all, z >. And we hae d 3 L3 (, if and only if set in ( L (,, y y,.the largest ompat inariant M {, } is the single ton { E }. Therefore, the endemi equilibrium E is globally asymptotially stable by the LaSalle inariane priniple [5]. The theorems are proed.
4 Zhengzhi Cao 3 Disussion Korobeinio [6-7] onstruted a lass of Lyapuno funtion. In [3], they proed global stability of the irus model with the inidene rate β p y q and they hae presented a global analysis of model (.) by Lyapuno funtions. In present paper, a lass of more general HIV- infetion model with Beddington-DeAngelis inidene rate and CTL immune response is onsidered. Referenes [] R. Anderson, R. Ma Infetious Diseases of Humans: Dynamis and Control, Oford Uniersity Press, 99. [] J.R. Beddington, Mutual interferene between parasites or predators and its effet on searhing effiien J. Anim.Eol. 44 (975), 33 34. [3] R. De Boer, A. Perelson, Target ell limited and immune ontrol models of HIV infetion: a omparison, Journal of Theoretial Biology, 9 (3) (998), 4. [4] D.L. DeAngelis, R.A. Goldstein, R.V. O Neill, A model for tropi interation. Eolog 56(975), 88 89. [5] J.K. Hale, Verduyn Lunel, Introdution to FuntionalDifferential Equations. Springer, New Yor, 993. [6] A. Korobeinio Global properties of basi irus dynamis models, Bull. Math. Biol, 66(4), 879-883. [7] A. Korobeinio P.K.A. Maini, Lyapuno funtion and global properties for SIR and SEIR epidemiologial models with nonlinear inidene, Math. Bios, (4), 57-6. [8] C. Long, H. Qi, S.H. Huang, Mathematial modeling of ytotoi lymphoyte-mediated immune responses to hepatitis B irus infetion. J. Biomed. Biotehnol.38(8), 573 585. [9] M. Nowa, C.R.M. Bangham, Population dynamis ofimmune responses to persistent iruses, Siene, 7(996), 74 79.
Global properties of an improed Hepatitis B irus model 5 [] M. Nowa, R. Ma Virus Dynamis, Oford Uniersity Press,. [] A. Perelson, P. Nelson, Mathematial models of HIV dynamis in io, SIAM Re 4(999), 3-44. [] A. Perelson, A. Neumann, M. Marowitz, J. Leonard, HIV- dynamis in io: ision learane rate, infeted ell life-span, and iral generation time. Siene, 7(996),58 586. [3] X. Wang, Y.D. Tao, Lyapuno funtion and global proper-ties OS irus dynamis with CTL immune response. Int. J. Biomath, 4(8), 443-448. Reeied: June, 4